Abstract

Laser speckle produced from a diffuse object can be used in determining the angular position of a rotating object. When the object rotates the backscattered speckle pattern, which changes continuously but repeats exactly with every revolution, is sampled by a suitably positioned photodetector. The photodetector output signal is periodic, and one period is stored in the memory as a reference. Shaft position can then be determined by the comparison of this stored reference signal with the current photodetector output signal. When the shaft is axially displaced, for example, by vibration, the backscattered speckle pattern changes on the photodetector and the similarity between the reference signal and the current signal is reduced. We examine the cross correlation of the real-time photodetector output signal and the stored reference signal as a function of axial shaft position. Use of a rotating shaft when collecting data is shown to be an efficient means by which to make effectively several thousand independent estimates of the maximum axial displacement tolerable before decorrelation of the photodetector output. Theoretical results and experiments conducted show that the decorrelation displacement varies, according to optical configuration, to a maximum value of 0.7 of the beam diameter. This has important implications for a proposed laser torquemeter as well as additional applications in which changes to the sampled speckle pattern, including decorrelation, are either desirable or undesirable.

© 1998 Optical Society of America

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References

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  1. T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]
  2. N. George, “Speckle from rough moving objects,” J. Opt. Soc. Am. 66, 1182–1194 (1976).
    [CrossRef]
  3. N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
    [CrossRef]
  4. T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
    [CrossRef]
  5. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]

1991 (1)

T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
[CrossRef]

1981 (2)

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
[CrossRef]

1976 (1)

Asakura, T.

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
[CrossRef]

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

George, N.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

Halliwell, N. A.

T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
[CrossRef]

Iwai, T.

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
[CrossRef]

Rothberg, S. J.

T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
[CrossRef]

Takai, N.

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
[CrossRef]

Wilmshurst, T. H.

T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
[CrossRef]

Appl. Phys. (1)

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Appl. Phys. B (Photophys. Laser Chem.) (1)

N. Takai, T. Iwai, T. Asakura, “An effect of curvature of rotating diffuse objects on the dynamics of speckles produced in the diffraction field,” Appl. Phys. B (Photophys. Laser Chem.) 26, 185–192 (1981).
[CrossRef]

Electron. Lett. (1)

T. H. Wilmshurst, S. J. Rothberg, N. A. Halliwell, “Laser torquemeter: a new instrument,” Electron. Lett. 27, 186–187 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the optical arrangement showing the coordinate axes.

Fig. 2
Fig. 2

Typical photodetector output.

Fig. 3
Fig. 3

Spectrum of photodetector output.

Fig. 4
Fig. 4

Gaussian approximation of the aperture autocorrelation function and the actual triangular autocorrelation function of the aperture.

Fig. 5
Fig. 5

Normalized autocorrelation as a function of axial displacement: (a) z = 30 cm, D = 1.45 mm, x c = 0.62 mm; (b) z = 16 cm, D = 1.18 mm, x c = 0.54 mm; (c) z = 57 cm, D = 1.45 mm, x c = 0.64 mm.

Fig. 6
Fig. 6

Plot of theoretical x c against experimental x c .

Fig. 7
Fig. 7

Plot of the normalized cross-correlation function for y/ D = 0, 0.1, and 0.2: D = 1 mm, z = 28 cm, L y = 1 mm.

Tables (1)

Tables Icon

Table 1 Summary of the Optical Configurations Showing the Effectiveness in Resisting Decorrelation of a Photodetector Output

Equations (18)

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Γ Δ I Δ x ,   τ = Δ I x ,   t Δ I x + Δ x ,   t + τ ,
Γ A X ,   τ = -   A x Δ I x ,   t d x -   A x + Δ x - X × Δ I x + Δ x ,   t + τ d Δ x ,
Γ A X ,   τ = - -   A x A x + Δ x - X d x × Δ I x ,   τ Δ I x + Δ x ,   t + τ d Δ x .
-   A x A x + Δ x - X d x = L x Λ Δ x - X L x L y Λ Δ y - Y L y L x exp - k 0 Δ x - X L x 2 L y exp - k 0 Δ y - Y L y 2 .
Γ Δ I Δ x ,   τ = c   exp - 4 | v | 2 τ 2 D 2 × exp - π D 2 λ z 2 | Δ x - ρ v τ | 2 ,
σ 0 = 2 2 π λ z D .
γ A X ,   Y ,   τ = Γ A X ,   Y ,   τ Γ A 0 ,   0 ,   0 = exp - 2 D 2 + 2 ρ σ 0 2 × ν ξ 2 + ν η 2 τ 2 exp - k 0 X L x 2 + Y L y 2 × exp k 0 σ 0 2 X + 2 ρ L x 2 ν ξ τ 2 L x 2 σ 0 2 k 0 σ 0 2 + 2 L x 2 × exp k 0 σ 0 2 Y + 2 ρ L y 2 ν η τ 2 L y 2 σ 0 2 k 0 σ 0 2 + 2 L y 2 .
θ c = 2 2 R D 2 + 2 k 0 2 z + ρ R 2 k 0 σ 0 2 + 2 L y 2 - 1 / 2
γ A τ = exp - 2 D 2 + 2 k 0 ρ 2 k 0 σ 0 2 + 2 L x 2 ν ξ 2 τ 2 .
x c D = k 0 σ 0 2 + 2 L x 2 2 k 0 σ 0 2 + 4 L x 2 + k 0 ρ 2 D 2 1 / 2 .
2 k 0 σ 0 2 + 4 L x 2     k 0 ρ 2 D 2 .
S = s 11 s 12 s 13 s 14 s 1 n s 1 N s 21 s 22 s 23 s 24 s 2 n s 2 n · · · · s mn · · s M 1 s M 2 s M 3 s M 4 · s MN .
Γ nn a = m = 1 M   s mn s m + a n .
Γ ¯ nn a = 1 N n = 1 N m = 1 M   s mn s m + a n .
Γ m m + a 0 = n = 1 N   s mn s m + a n .
Γ ¯ m m + a 0 = 1 M m = 1 M n = 1 N   s mn s m + a n .
Γ ¯ nn a = M N   Γ ¯ m m + a 0 ,
γ ¯ nn a = Γ ¯ nn a Γ ¯ nn 0 = Γ ¯ m m + a 0 Γ ¯ mm 0 .

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